Simplify/Condense log base 6 of 1/( square root of 6)

$\log_{6} (\frac{1}{\sqrt{6}})$

Step 1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\log_{6} (\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 2

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\log_{6} (\frac{\sqrt{6}}{\sqrt{6} \sqrt{6}})$

Raise $\sqrt{6}$ to the power of $1$ .

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}})$

Raise $\sqrt{6}$ to the power of $1$ .

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{6} (\frac{\sqrt{6}}{{\sqrt{6}}^{2}})$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\log_{6} (\frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{6} (\frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}})$

Combine $\frac{1}{2}$ and $2$ .

$\log_{6} (\frac{\sqrt{6}}{6^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{6} (\frac{\sqrt{6}}{6^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{6} (\frac{\sqrt{6}}{6^{1}})$

Evaluate the exponent.

$\log_{6} (\frac{\sqrt{6}}{6})$

Step 3

Rewrite $\frac{\sqrt{6}}{6}$ as $\sqrt{6} \cdot 6^{- 1}$ .

$\log_{6} (\sqrt{6} \cdot 6^{- 1})$

Step 4

Rewrite $\log_{6} (\sqrt{6} \cdot 6^{- 1})$ as $\log_{6} (\sqrt{6}) + \log_{6} (6^{- 1})$ .

$\log_{6} (\sqrt{6}) + \log_{6} (6^{- 1})$

Step 5

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{6} (\sqrt{6}) - \log_{6} (6)$

Step 6

Logarithm base $6$ of $6$ is $1$ .

$\log_{6} (\sqrt{6}) - 1 \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$\log_{6} (\sqrt{6}) - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\log_{6} (\sqrt{6}) - 1$

Decimal Form:

$- 0.5$

Simplify/Condense log base 6 of 1/( square root of 6)

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